1.4.5 Gamma

Gamma is the second derivative of the option price with respect to spot price—or the first derivative of delta with respect to spot—representing the rate at which delta fluctuates as the price of the underlying varies. Gamma is vital to the delta neutral portfolio, as delta must be dynamically hedged as the price of the underlying moves. From the perspective of the market maker, options sold to market participants result in negative gamma, and options bought from market participants result in positive gamma. The consequence for a delta neutral market maker is that contracts sold to them cause hedging that resists movement in spot price, and contracts bought from them cause hedging that amplifies movement in price.

Example

A market participant sold 12 SPY 600c 45 day expiry when the SPY was trading at $595 for $7.00 per contract with a delta of 0.7, and created a delta neutral position by buying 840 shares of the SPY. The options have a gamma of 0.025 delta/dollar at this price, and the price of SPY moves up $4 to $599. To maintain delta neutral positioning, the market participant must buy 120 additional SPY shares.

Summary of Terms

$V$ = Contract Value

$\Gamma$ = Contract Gamma

$\Delta$ = Contract Delta

$S$ = Spot Price

$K$ = Strike Price

$\sigma$ = Implied Volatility

$\tau$ = Years to Expiration

$r$ = Risk Free Rate

$q$ = dividend yield

Calculation

\Gamma = \frac{\partial^2 V}{\partial S^2} = \frac{\partial \Delta}{\partial S}

\Gamma = {1 \over S\sigma\sqrt{2\pi\tau}}e^{-{1 \over 2}d_+^2}

d_+ = {1 \over \sigma \sqrt{\tau}}\bigg[\ln\bigg({S \over K}\bigg) + \bigg(r - q + {\sigma^2 \over 2}\bigg)\tau\bigg]

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